Changes

The '''Taylor series''' of a function is useful for approximating a [[function|mathematical function]] near to some particular point. For a function <math>f(x)</math>, the Taylor series about the point <math>x_0</math> is

The '''Taylor series''' of a function is useful for approximating a [[function|mathematical function]] near to some particular point. For a function <math>f(x)</math>, the Taylor series about the point <math>x_0</math> is

<math>f(x)=f(x_0)+(x-x_0)\frac{df(x_0)}{dx}+\frac{(x-x_0)^2}{2!}\frac{d^2f(x_0)}{dx^2}+\ldots+\frac{(x-x_0)^N}{N!}\frac{d^Nf(x_0)}{dx^N}</math>

<math>f(x)=f(x_0)+(x-x_0)\frac{df(x)}{dx}+\frac{(x-x_0)^2}{2!}\frac{d^2f(x)}{dx^2}+\ldots+\frac{(x-x_0)^N}{N!}\frac{d^Nf(x)}{dx^N}</math>

where each of the derivatives is to be evaluated at <math>x=x_0</math>. If as <math>N\rightarrow\infty</math> the [[series (mathematics)|series]] converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around <math>x_0=0</math>, in which case they are sometimes also known as a [[Maclaurin series]].

where each of the derivatives is to be evaluated at <math>x=x_0</math>. If as <math>N\rightarrow\infty</math> the [[series (mathematics)|series]] converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around <math>x_0=0</math>, in which case they are sometimes also known as a [[Maclaurin series]].